2016
DOI: 10.4310/mrl.2016.v23.n5.a5
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$S^1$-equivariant local index and transverse index for non-compact symplectic manifolds

Abstract: We define an S 1 -equivariant index for non-compact symplectic manifolds with Hamiltonian S 1 -action. We use the perturbation by Dirac-type operator along the S 1 -orbits. We give a formulation and a proof of quantization conjecture for this S 1 -equivariant index. We also give comments on the relation between our S 1equivariant index and the index of transverse elliptic operators.2010 Mathematics Subject Classification. Primary 53D50 ; Secondary 19K56, 58J20.

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Cited by 3 publications
(11 citation statements)
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“…The conjecture was proved by Ma and Zhang [8], and Paradan [12] gave a new proof of it. In [3] the author gave a formulation of a variant of S 1 -equivariant index for non-compact symplectic manifold in the context of this paper. The S 1 -equivariant index coincides with the transverse index for proper moment map without critical points.…”
Section: Localization Of Equivariant Riemann-roch Numbersmentioning
confidence: 99%
See 1 more Smart Citation
“…The conjecture was proved by Ma and Zhang [8], and Paradan [12] gave a new proof of it. In [3] the author gave a formulation of a variant of S 1 -equivariant index for non-compact symplectic manifold in the context of this paper. The S 1 -equivariant index coincides with the transverse index for proper moment map without critical points.…”
Section: Localization Of Equivariant Riemann-roch Numbersmentioning
confidence: 99%
“…If the moment map is not proper, then, they however do not coincide in general. See [3] for details.…”
Section: Localization Of Equivariant Riemann-roch Numbersmentioning
confidence: 99%
“…Moreover one can see thatˆis transversally elliptic in the sense of Atiyah [3]. In fact for any -invariant function ℎ : → R, since commutes with the multiplication by ℎ one hasˆℎ − ℎˆ= ℎ − ℎ = ( ℎ), 7 The third condition implies that ( 2 ) ( ) is a strictly positive operator on each -orbit. On the other hand as in Lemma 2.1 the condition (c) implies that + is also a differential operator along the orbits, we can take such a constant for each orbit.…”
Section: Deformation Along Orbits and Quantization Of Non-compact Manmentioning
confidence: 96%
“…3 gives an alternative proof of Vergne's conjecture for torus case to Ma-Zhang's proof in [20] which uses Braverman's deformation. (3) The above construction and a proof of Theorem 5.3 is essentially same as those in [7].…”
Section: [Qr]=0 For Non-compact Hamiltonian Torus Manifoldsmentioning
confidence: 99%
“…These data define a Fredholm operator on L 2 ( W ) (ρ) as in Corollary 2.4. Though we agree that it is a little bit strange notation 10 , we denote this index by…”
Section: Deformation Along Orbits and Quantization Of Non-compact Man...mentioning
confidence: 99%